We study the elliptic system \begin{equation*} \begin{cases} -\Delta u_1 - \kappa_1u_1 = \mu_1|u_1|^{p-2}u_1 + \lambda\alpha|u_1|^{\alpha-2}|u_2|^\beta u_1, \\ -\Delta u_2 - \kappa_2u_2 = \mu_2|u_2|^{p-2}u_2 + \lambda\beta|u_1|^\alpha|u_2|^{\beta-2}u_2, \\ u_1,u_2\in D^{1,2}_0(\Omega), \end{cases} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb R^N$, $N\geq 3$, $\kappa_1,\kappa_2\in\mathbb R$, $\mu_1,\mu_2,\lambda> 0$, $\alpha,\beta> 1$, and $\alpha + \beta = p\le 2^*:={2N}/({N-2})$. For $p\in (2,2^*)$ we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for $\lambda$ sufficiently large. If $p=2^*$ and $\kappa_1,\kappa_2> 0$ we establish the existence of a ground state for $\lambda$ sufficiently large if, either $N\ge5$, or $N=4$ and neither $\kappa_1$ nor $\kappa_2$ are Dirichlet eigenvalues of $-\Delta$ in $\Omega$.

This paper deals with a nonlinear Volterra delay evolution inclusion subjected to a nonlocal implicit initial condition. The evolution inclusion involves an $m$-dissipative operator (possibly multivalued and/or nonlinear) and a noncompact interval. We first consider the evolution inclusion subjected to a local initial condition and prove an existence result for bounded $C^0$-solutions. Then, using a fixed point theorem for upper semicontinuous multifunctions with contractible values, we obtain a global solvability result for the original problem. Finally, we present an example to illustrate the abstract result.

Existence and spatio-temporal patterns of periodic solutions to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer $O(2) \times \Gamma \times \mathbb Z_2$-equivariant degree theory, where $O(2)$ is related to the reversing symmetry, $\Gamma$ reflects the symmetric character of the coupling in the corresponding network and $\mathbb Z_2$ is related to the oddness of the right-hand side. Abstract results are supported by a concrete example with $\Gamma = D_6$ - the dihedral group of order 12.

We consider the following nonlinear elliptic systems: \begin{equation*} \left\{\begin{alignedat}2 -\Delta u_1 +\lambda_1u_1 & = \mu_1u_1^3+\beta_{12}u_1u_2^2+\beta_{13}u_1u_3^2 &\quad& \hbox{in } \Omega,\\ -\Delta u_2 +\lambda_2u_2 & = \mu_2u_2^3+\beta_{12}u_2u_1^2+\beta_{23}u_2u_3^2 &\quad &\hbox{in } \Omega,\\ -\Delta u_3 +\lambda_3u_3 & = \mu_3u_3^3+\beta_{13}u_3u_1^2+\beta_{23}u_3u_2^2 &\quad& \hbox{in } \Omega,\\ \overrightarrow{u} & =(u_1,u_2,u_3)\in H^1_0(\Omega)^3, \end{alignedat}\right. \end{equation*} where $\Omega\subset \R^n$ ($n\leq 3$) is a bounded domain with smooth boundary, $\lambda_j, \mu_j> 0$ $(j=1,2,3)$, $\beta_{12}> 0$, and $\beta_{13}, \beta_{23}\leq 0$. For this model case of coupled Schrödinger equations with mixed coupling, for sufficiently large $\beta_{12}> 0$, we show that there exists a solution $(u_{1\beta},u_{2\beta},u_{3\beta})$ of $(*)$ such that $u_{1\beta}> 0, u_{2\beta}> 0$ and $u_{3\beta}$ changes sign exactly once. We also show that, for any given $k\in \mathbb{N}$, there exist $k$ vector solutions $(u_{1\beta}^\ell,u_{2\beta}^\ell,u_{3\beta}^\ell)$ $(\ell=1,\ldots,k)$ and these solutions are characterized by the genus with respect to a partial symmetry $\sigma(u_1,u_2,u_3)=(-u_1,-u_2,u_3)$.

We show that the typical nonexpansive mapping on a small enough subset of a CAT($\kappa$)-space is a contraction in the sense of Rakotch. By typical we mean that the set of nonexpansive mapppings without this property is a $\sigma$-porous set and therefore also of the first Baire category. Moreover, we exhibit metric spaces where strict contractions are not dense in the space of nonexpansive mappings. In some of these cases we show that all continuous self-mappings have a fixed point nevertheless.

A compact polyhedron $X$ is said to have the Bounded Index Property for Homotopy Equivalences (BIPHE) if there is a finite bound $\B$ such that for any homotopy equivalence $f\colon X\rightarrow X$ and any fixed point class $\mathbf{F}$ of $f$, the index $|\mathrm{ind}(f,\mathbf{F})|\leq \mathcal{B}$. In this note, we consider the product of compact polyhedra, and give some sufficient conditions for it to have BIPHE. Moreover, we show that products of closed Riemannian manifolds with negative sectional curvature, in particular hyperbolic manifolds, have BIPHE, which gives an affirmative answer to a special case of a question asked by Boju Jiang.

This paper presents a generalized epidemic model with discontinuous treatment strategies and time-varying delays. Under the concept of Filippov solution, by applying the differential inclusions and the topological degree theory in set-valued analysis, we employ a novel argument to establish new results on the existence of the periodic solutions for the considered epidemic model. After that, we derive some criteria on the uniqueness, global exponential stability of the considered epidemic model and convergence of the corresponding autonomous case of the considered epidemic model, in terms of nonsmooth analysis theory with the Lyapunov-like approach. Our results extend previous works on the epidemic model to the discontinuous cases, some corresponding results in the literature can be enriched and extended. Finally, typical examples and the corresponding numerical simulations have been carried out to support the analytic findings.

We deal with an aggregate version of the KKM theorem. Given $n$ KKM families of special type on an $( n-1)$-dimensional simplex, we show that it is possible to choose a single element from every KKM family to get a KKM family on that simplex. We also introduce and study function KKM families as surrogates of KKM families on simplexes. We show a function version of the KKM theorem. The Coincidence Theorem is our main result, Brouwer's fixed is a special cases of that theorem.

We will show that, unlike to usual (i.e. non-impulsive) differential equations, the Nielsen theory results for single-valued as well as multivalued maps on tori can be effectively applied to impulsive differential equations and inclusions. With this respect, two main aims will be focused, namely: (i) multiplicity results for harmonic periodic solutions, (ii) the coexistence of subharmonic periodic solutions with various periods. In both cases, we will try to contribute at least partly to the problem posed already in 1950 by Jean Leray. A dynamic complexity of the related maps, measured in terms of entropy, will be also examined.

In this paper we prove the existence of multiple periodic (harmonic and subharmonic) solutions for a class of planar Hamiltonian systems which includes the case of the second order scalar ODE $x'' + a(t)g(x) = 0$ with $g$ satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincare-Birkhoff fixed point theorem as well as some refinements on the side of the theory of topological horseshoes. A Duffing-type equation and an exponential nonlinearity case are studied as applications.